I think I have a way to do it, but read my note at the bottom about why you may not find it to be valid.

Use an eye-shaped figure, but where the ends of the eyes meet at angle 0. This will allow unique tangents at the "corners".

     ---
/     \
--         --
\     /
---


It's kind of hard to draw, but the

The eyes will fit together in a standard brick pattern. If you make

In the eyes from image, I used a sin function, but you can do it also using circle fragments. In this case, then the tiling corresponds to a standard penny tiling of the plane, but apportioning the empty space to adjacent pennies in order to make the eye shapes.

Since my Jordan curves turn directly around in the opposite direction at those cusps, however, you may not consider this to be a valid example, since perhaps you regard this as two tangent lines at those points, pointing in opposite directions.

I think someone will show up and prove that you cannot do it without any cusps. I would like to know whether you can do it with only one cusp.

4 deleted 341 characters in body; edited body

I think I have a way to do it, but read my note at the bottom about why you may not find it to be valid.

Use an eye-shaped figure, but where the ends of the eyes meet at angle 0. This will allow unique tangents at the "corners".

     ---
/     \
--         --
\     /
---


It's kind of hard to draw, but the eyes will fit together in a standard brick pattern. If you make the eyes from circle fragments, then the tiling corresponds to a standard penny tiling of the plane, but apportioning the empty space to adjacent pennies in order to make the eye shapes.

Since my Jordan curves turn directly around in the opposite direction at those cusps, however, you may not consider this to be a valid example, since perhaps you regard this as two tangent lines at those points, pointing in opposite directions.

Here is another way to do it, using only one cusp. Use a balloon or sperm-shaped figure, built from circle fragments:

              ____
/      \
|        |
\      /
\  /
V
|



These shapes can be placed directly adjacent to each other, with the next row inverted. This is the tiling obtained from a standard circle packing, but giving the space between the circle to the circle directly above or below.

Lastly,

I think someone will show up and prove that you cannot do it without any cusps.

3 added 624 characters in body

I think I have a way to do it, but read my note at the bottom about why you may not find it to be valid.

Use an eye-shaped figure, but where the ends of the eyes meet at angle 0. This will allow unique tangents at the "corners".

     ---
/     \
--         --
\     /
---


It's kind of hard to draw, but the eyes will fit together in a standard brick pattern.

Since my Jordan curves turn directly around in the opposite direction at those cusps, however, you may not consider this to be a valid example, since perhaps you regard this as two tangent lines at those points, pointing in opposite directions.

Here is another way to do it, using only one cusp. Use a balloon or sperm-shaped figure, built from circle fragments:

              ____
/      \
|        |
\      /
\  /
V
|



These shapes can be placed directly adjacent to each other, with the next row inverted. This is the tiling obtained from a standard circle packing, but giving the space between the circle to the circle directly above or below.

Lastly, I think someone will show up and prove that you cannot do it without any cusps.

2 deleted 718 characters in body
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